# A ball with a mass of #3# #kg # and velocity of #1 # #ms^-1# collides with a second ball with a mass of #5# #kg# and velocity of #- 8# #ms^-1#. If #25%# of the kinetic energy is lost, what are the final velocities of the balls?

The equations yield two sets of answers:

OR

Momentum is conserved in all collisions. In this instance, kinetic energy is not conserved: 25% is lost and 75% remains.

The initial momentum is given by:

The final momentum will be the same as the initial momentum.

(if we define 'to the right' as the positive direction, the net momentum of the system is to the left)

The initial kinetic energy is given by:

The final kinetic energy will be 75% of the initial kinetic energy:

The expression for the final momentum is:

The expression for the final kinetic energy is:

We now have two equations in two unknowns, so we can solve the system.

Divide Equation 1 by 3 and rearrange to give:

Rearranging:

We can solve this quadratic equation using the quadratic formula (or other methods):

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The final velocity of the 3 kg ball is ( -1 , \text{m/s} ), and the final velocity of the 5 kg ball is ( -3 , \text{m/s} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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